weak consistency of the euler method for numerically solving stochastic differential equations with...

Stochastic Processes and their Applications 76 (1998) 33–44

Weak consistency of the Euler method fornumerically solving stochastic di�erential equations

with discontinuous coe�cients

K.S. Chan ∗, O. StramerDepartment of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA

Received 28 April 1997; accepted 2 March 1998

Abstract

We prove that, under appropriate conditions, the sequence of approximate solutions con-structed according to the Euler scheme converges weakly to the (unique) solution of a stochasticdi�erential equation with discontinuous coe�cients. We also obtain a su�cient condition forthe existence of a solution to a stochastic di�erential equation with discontinuous coe�cients.These results are then applied to justify the technique of simulating continuous-time thresholdautoregressive moving-average processes via the Euler scheme. c© 1998 Elsevier Science B.V.All rights reserved.

AMS classi�cation: 62F17; 60J55; 60H10

Keywords: Good integrators; Martingale di�erences; Threshold ARMA processes

1. Introduction

In an approach to modeling irregularly sampled time series, it is assumed that thereis an underlying continuous-time process from which the discrete-time data is obtained.The underlying continuous-time process is often modeled by the following stochasticdi�erential equation:

dX (t)= b(X (t)) dt + �(X (t)) dB(t); (1)

or equivalently in coordinate form,

dXi(t)= bi(X (t)) dt +r∑j=1

�ij(X (t)) dBj(t); 16i6d; (2)

where B=(B1; : : : ; Br) is an r-dimensional Brownian motion (r¿1) starting from theorigin, and b :Rd→Rd (the drift vector) and a := ��T :Rd→Rd ⊗ Rd (the di�usionmatrix) are locally bounded Borel measurable functions. With no loss of generality,we assume that the initial condition is that X0 = x0, a �xed vector.

∗ Corresponding author.

0304-4149/98/$19.00 c© 1998 Elsevier Science B.V. All rights reservedPII: S0304 -4149(98)00020 -9

34 K.S. Chan, O. Stramer / Stochastic Processes and their Applications 76 (1998) 33–44

Recently, nonlinear continuous-time models have received much attention in theliterature, see Brockwell (1993), Tong and Yeung (1991) and Ozaki (1985). A classof useful nonlinear continuous-time models is the Threshold Autoregressive MovingAverage (CTARMA) model for which the drift is piecewise linear and the di�usionterm is piecewise constant. See Section 3 for the de�nition of the CTARMA models.For the CTARMA models, the drift and the di�usion terms are usually discontinuous.The classical result on the uniqueness and the existence of a solution to Eq. (1) requiresthat the coe�cients be Lipschitz continuous, and hence it is not applicable for theCTARMA models. See, however, Nisio (1973), Stramer et al. (1996a,b) and Brockwelland Williams (1995) for some results on the existence and uniqueness of CTARMAmodels of lower order. Motivated by the CTARMA models, we consider the problem ofnumerically solving the stochastic di�erential equation (1) with possibly discontinuousdrift and di�usion terms.The Euler scheme is a general method for numerically solving Eq. (1) by “discretiz-

ing” the process to a stochastic di�erence equation:

Xn((k + 1)=n) =Xn(k=n) + b(Xn(k=n))=n

+�(Xn(k=n))�k+1=√n; k =0; 1; : : : ; n− 1

(3)

where {�k =(�1k ; : : : ; �rk)} is a sequence of i.i.d. random vectors with i.i.d. componentsof zero mean and unit variance. (With no loss of generality, we assume that it isrequired to simulate {X (t); 06t61} satisfying Eq. (1).) The discrete-time processcan then be extended to the unit interval by de�ning Xn(t)=Xn([t=n]) where [x] de-notes the integral part of x. Our main result, Theorem 2.1, shows that assuming thatEq. (1) admits a unique solution, denoted by X , and under suitable regularity condi-tions on the set of discontinuity for the coe�cients, Xn→X weakly. This generalizesthe weak convergence result for the Euler scheme when the coe�cients are continuous.See Du�e and Protter (1992), Kurtz and Protter (1991a) and Kloeden and Eckhard(1992). We also show in Proposition 2.5 that subject to linear growth conditions onthe drift and the di�usion terms, and assuming suitable regularity conditions on theset of discontinuity for the coe�cients, there exists a (not necessarily unique) solutionto Eq. (1). In Proposition 2.6, we show that under suitable conditions, the momentsof the Euler scheme converge to those of the unique solution X . In Section 3, weapply the general results in Section 2 to study the validity of the Euler scheme for theCTARMA models. We also show that the de�ning stochastic di�erential equation of aCTARMA model always admits a solution, but the uniqueness of the solution is stillan open problem for the higher order CTARMA models. We conclude in Section 4 bybrie y mentioning some future research problems.

2. Main results

Unless stated otherwise, we assume that X is the unique (in law) Rd valuedcontinuous-time process satisfying Eq. (1), where b(·) and �(·) are locally boundedmeasurable functions. Without loss of generality (WLG), we set the initial conditionto be X0 = x0, where x0 is a d-dimensional vector of �xed numbers. We shall also

K.S. Chan, O. Stramer / Stochastic Processes and their Applications 76 (1998) 33–44 35

assume that there exist �nitely many pairwise disjoint sets O1; : : : ; Ok ⊆Rd for whichthe following regularity conditions hold.(1) Rd=

⋃ki=1 Oi, and over each Oi; b(·) and �(·) are continuous. For each i, the re-

striction of b (and that of �) over Oi can be extended to an everywhere continuousfunction.

(2) For each r; ∃ an integer n(r) such that the interior of Or ,

interior(Or)=n(r)⋂‘=1

{x∈Rd; fr‘(x)¿0};

where fr‘ ∈C∞(Rd); ∀‘; ∀r.Consider the discrete-time approximation Xn de�ned by Eq. (3). Its extension over

[0; 1] satis�es the following equation:

Xn(t)= x0 +∫ t

0b(Xn([sn]=n−) d�n(s) +

∫ t

0�(Xn([sn]=n−) dBn(s); (4)

where Xn(s−) denotes the left limit limt¡s; t→s Xn(t); Xn(0−)= x0; �n(s)= [sn]=n;Bn(s)=(B1n(s); : : : ; B

rn(s)) where for all 16i6r; B

in(s)=

∑[sn]k=1 �

ik =√n and {{�ik}ri=1}∞k=1

are i.i.d. with zero mean and of unit variance (Bin(s) is de�ned to be zero if there areno summands in the summation). Here the superscript i denotes the ith component;only in a few occasions shall we use the superscript to denote taking power, but thecontext should render the meaning of the superscript clear. Note that Xn(t) is identicallyequal to Xn(k=n) over [k=n; (k + 1)=n).It follows from the invariance principle that Bn converges weakly to the Brownian

motion B in D[0; 1]r under the topology of uniform convergence; see Billingsley (1968).Also, �n(t) converges to t uniformly over [0; 1]. This suggests that Xn converges weaklyto a solution of Eq. (1). The following result says that this is indeed true undersuitable regularity conditions, the proof of which is inspired by the related “patchworkmartingale problem” studied by Kurtz (1990).

Theorem 2.1. Suppose that Eq. (1) admits a unique continuous solution, denoted byX. Let fr‘ , be de�ned as in the Regularity Conditions and assume (A1): ∀r; ∀‘;

∑di=1∑d

k=1 aik(@fr‘ =@xi) (@fr‘ =@xk) is bounded below from zero over any compact subsetsof some neighborhood of �=

⋃i @Oi, the union of the boundaries of the O’s. Then,

Xn de�ned by (4) converges weakly to X as n→∞.

Proof of Theorem 2.1. WLG, assume that the state space Rd=O1 ∪ O2, where thecommon boundary @O1 = @O2 = {o: f(o)= 0}. Let bi(�i) be an everywhere contin-uous function which coincide with b(�) over Oi. For notational convenience, de-�ne X̃ n(s)=Xn([sn]=n−). De�ne the occupation measure of Xn on Oi by �in(t)=∫ t0 5Oi(X̃ n(s)) ds; ∀t¿0: Note that �in(s) are continuous non-decreasing functions ins. Clearly, we have the following equalities:(a) �in(t)=

∫ t0 5Oi(X̃ n(s)) d�in(s); ∀t¿0:

(b)∑2

i=1 �in(t)= t; ∀t¿0:


Let {�in}; i=1; 2 be two independent sequences of random variables having the samedistribution as that of {�n}. De�ne Bin(t)=

∑[tn]k=1 �ik =

√n. Because {�in} are i.i.d., Xn

de�ned by Eq. (4) is equivalent to

Xn(t)= x0 +2∑i=1

{∫ t

0bi(X̃ n(s)) d�in(s) +

∫ t

0�i(X̃ n(s)) dBin ◦ �in(s)

}: (5)

Because bi and �i are globally bounded (otherwise we will apply suitable stoppingtime arguments), (Xn; �in; Bin; Bin◦�in; i=1; 2) are tight in the product Skorohod spaces.Moreover, (�in; Bin; Bin◦�in) are “good” integrators, whose concept we now brie y sum-marize; see Kurtz and Protter (1991a), Du�e and Protter (1992) for details. Let Yn bean Rm-valued cadlag process, and Hn be a k×m real valued cadlag matrix process, i.e.,all their component processes are right-continuous functions with left-hand limits thatare de�ned on [0; T ], where T is a �xed positive number. A sequence {Yn} of semi-martingales is de�ned to be good if for any {Hn}, the weak convergence of {(Hn; Yn)}to (H; Y ) implies that (i) Y is a semimartingale and (ii) (Hn; Yn;

∫ · Hn(s−) dYn(s))converges weakly to (H; Y;

∫ · H (s−) dY (s)).To prove the theorem, it su�ces to show that the limit of any (weakly) convergent

subsequence of (Xn) satis�es Eq. (1). WLG, assume then that (Xn; �in;Bin; Bin ◦ �in; i=1; 2) converges to (X; �i; Bi; B̃i; i=1; 2) a.s., as n→∞. Clearly, Biare two independent Brownian motions whose covariance at t equals tI , where I is ther × r identity matrix. Also, it could be veri�ed that B̃i=Bi ◦ �i. The good integratorproperty of (�in; Bin; Bin ◦ �in) alluded to above implies that

X (t)= x0 +2∑i=1

{∫ t

0bi(X (s)) d�i(s) +

∫ t

0�i(X (s)) dBi ◦ �i(s)

}: (6)

We need to show that the preceding equation is equivalent to Eq. (1). This will bedone by making use of the following lemma, whose proof will be given after the proofof this theorem.

Lemma 2.2. Assuming that the conditions of Theorem 2:1 hold; then ∀t¿0(c)

∑2i=1 �i(t)= t;

(d) �i(t)=∫ t0 5Oi(X (s)) d�i(s).

Taken together, these two identities mean that �i( ) is the occupation measure of Xon Oi; i=1; 2: From (c) and (d), we get

∫ t

05Oj (X (s)) ds=

2∑i=1

∫5Oj (X (s)) d�i(s)

=2∑i=1

∫5Oj (X (s)5Oi(X (s)) d�i(s)

=∫ t

05Oj (X (s)) d�j(s):


De�ne B(t)=∑2

i= 1 Bi ◦ �i(t). Then, B is a Brownian motion whose covariance attime t equals tIr×r . This can be seen by noting that each component of B is acontinuous martingale, and for any �∈Rr , the quadratic variation of �TB(t)� equals�T�

∑�i(t)= t�T�. For any �∈Rp, the quadratic variation of

∫ t0 5Oj (Xs)�

TdBi ◦ �i(s)equals �T�

∫ t0 5Oj (X (s)) d�i(s)= �

T��ij�i(t) where � is the Kronecker delta. Therefore,∫ t0 5Oj (Xs)�

TdBi ◦ �i(s)= 0; ∀i 6= j. Hence,∫ t

05Oj (X (s)) dB(s) =

2∑i=1

∫ t

05Oj (X (s)) dBi ◦ �i(s)

=∫5Oj (X (s)) dBj ◦ �j(s)

=∫dBj ◦ �j(s);

because �j(s) increases i� X (s)∈Oj, according to the preceding lemma. Hence,

X (t) = x0 +2∑i=1

{∫ t

0bi(X (s)) d�i(s) +

∫ t

0�i(X (s)) dBi ◦ �i(s)

}

= x0 +2∑i=1

{∫ t

0bi(X (s)5Oi(X (s)) ds+

∫ t

0�i(X (s))5Oi(X (s)) dB(s)

}

= x0 +∫ t

0b(X (s)) ds+

∫ t

0�(X (s)) dB(s):

This completes the proof of the theorem.It remains to prove Lemma 2.2. Eq. (c) follows from (b) by taking limits. For (d),

we �rst verify that ∀i; ∀t;

�i(t)=∫ t

05 �Oi(X (s)) d�i(s): (7)

First, note that

�i(t)6∫ t

05 �Oi(X (s)) d�i(s); ∀t; ∀i: (8)

To see why this is true, let hm( ) be a sequence of uniformly continuous function that ↓5 �Oi( ). Furthermore, it is assumed that hm are globally bounded. Then, �in(t)=

∫ t0 5 �Oi(X̃ n

(s)) d�in(s)6∫ t0 hm(X̃ (s)) d�in(s), with the last term converging to

∫ t0 hm(X (s)) d�i(s),

and hence Eq. (8). However, for all t,

t=2∑i=1

�i(t)6∑i

∫ t

05 �Oi(X (s)) d�i(s)6

∑∫ t

0d�i(s)= t:

Therefore, for all i, �i(t)=∫ t0 5 �Oi(X (s)) d�i(s); ∀t¿0:

Below, X is assumed to be globally bounded by a constant, say K¿0. (Other-wise, we will apply suitable stopping time arguments.) Let g be a bounded smooth


function whose second derivative is a smooth approximation of 5[−�; �](x). More specif-ically let g(y)=

∫ y0 (∫ w0 g

′′(u) du) dw and g′′(y) is an even function, g′′(y)= 1 for06y6�, g′′(y)= 0 for 2�6y6K and g′′(y) is non-increasing for �¡y¡2� whereK¿1¿2�¿0. Note in particular that for |y|6K; |g′(y)|62� and |g(y)|62�K . Wealso assume that g′′(y) is de�ned for |y|¿K such that |g(y)|64�K and |g′(y)|62�for all y. Recall that the solution set of f=0, that is, {o: f(o)= 0} de�nes theboundary @O1 = @O2. Using Ito’s formula, we get (where Df or f′ denotes the �rstderivatives and D2f the matrix of second partial derivatives)

df(X (t))=DfT(X (t)) dX (t) +∑i

�Ti (X (t))D2f(X (t))�i(X (t)) d�i(t)=2;

dg ◦ f(X (t)) = g′(f(X (t)) df(X (t))

+g′′(f(X (t))

∑i

(DTf�i�Ti Df)(X (t)) d�i(t)=2:

The properties of g stated above then imply that there exist two positive constants Mand �, independent of �, such that for �¿0 su�ciently small,

M�¿ E{∣∣∣∣∫ t

0dg ◦ f(X (s))

∣∣∣∣+∣∣∣∣∫ t

0g′(f(X (s)) df(X (s))

∣∣∣∣}

¿ E

∣∣∣∣∣∫ t

0g′′(f(X (s))

∑i

(DTf�i�Ti Df)(X (s)) d�i(s)=2

∣∣∣∣∣¿ E

∫ t

05[−�; �](f(X (s))

∑i

(DTf�i�Ti Df)(X (s)) d�i(s)=2

¿ �E∫ t

05[−�; �](f(X (s)) ds;

the last equality being a consequence of the fact that∑

i(DTf�i�Ti Df) is bounded

below from 0 over any compact subsets of some neighborhood of �=⋃i @Oi. We are

now ready to prove (d) of Lemma 2.2 as follows:

E∣∣∣∣�i(t)−

∫ t

05Oi(X (s)) d�i(s)

∣∣∣∣ = E∫ t

05 �Oi\Oi(X (s)) d�i(s)

6 E∫ 1

05[−�; �](f(X (s)) ds6M�:

As �¿0 is arbitrary, �i(t)=∫ t0 5Oi(X (s)) d�i(s) a.s. It follows from the continuity of

�i( ) that it is almost sure that for all t¿0, �i(t)=∫ t0 5Oi(X (s)) d�i(s). This completes

the proof of Lemma 2.2.

Remark 2.3. As can be seen from the proof of Theorem 2.1, conditions (A1) may bereplaced by the more general condition (A2): ∃ a sequence of neighborhoods {�m}


shrinking towards � and ∀ compact set K; ∀T¿0

limmlim sup

nE(∫ T

05(Xn(t)∈K ∩�m) dt

)=0:

Remark 2.4. Theorem 2.1 can be partially extended to the case when the components ofthe random noise {�k}k¿1 are independent, ∀k, and each component of {�k}k¿1 form asequence of martingale di�erences. (For notational convenience, we write the i th com-ponent of �k as �k(i).) That is, ∀i, �k(i)= Sk(i)−Sk−1(i), where {Sk(i);Fk(i)} is a mar-tingale with S0(i)= 0 a.s., and Fk(i) is the � �eld generated by S0(i); S1(i); : : : ; Sk(i).Moreover, it is assumed that1. s−2n (i)

∑nk=1 �

2k(i)→ 1 and s−2n (i) supk6n �

2k(i)→ 0 in probability as n→∞, where

s2n(k)=ES2n (k).

2. (A1) holds or in the case of non-Gaussian �k , assume (A3): for each j=1; : : : ; r;∑di=1

∑dk=1 �ij�kj(@fr‘ =@xi)(@fr‘ =@xk) is bounded below from zero over any compact

subsets of some neighborhood of �.Because the equivalence of Eqs. (4) and (5) holds only under the i.i.d. assumption onthe noise sequence, the techniques employed for proving this extension are di�erentfrom that of Theorem 2.1; see Chan and Stramer (1996) for a proof.

We next show that if, in addition to the assumption in Theorem 2.1, we assumelinear growth conditions on the coe�cients b and �, then there exists a (not necessarilyunique) solution to Eq. (1).

Proposition 2.5. Assume that there exists R¿0 such that for all i=1; : : : ; d andj=1; : : : ; r; b2i (x) + �

2ij(x)6R(1 + ‖x‖2), for all x∈Rd. Let fr‘ be de�ned as in

the regularity conditions and assume (A1): ∀r; ∀‘; ∑di=1

∑dk=1 aik(@fr‘ =@xi)(@fr‘ =@xk)

is bounded below from zero over any compact subsets of some neighborhood of�=

⋃i @Oi, the union of the boundaries of the O’s. Then; there exists a solution

to (1). (Conditions (A1) may be replaced by condition (A2) stated in Remark 2.3.)

Proof. We �rst note that from Example 5.3.15 of Karatzas and Shreve (1991) wehave that Xn de�ned by Eq. (4) is tight and hence we can assume, without lossof generality that Xn de�ned by Eq. (4) converges weakly to some X as n→∞.Hence Xn(· ∧ �kn)→X k(·) :=X (·∧�k), where k¿0 is a �xed number and �kn is thestopping time when |Xn| �rst exceeds k and �k is the corresponding stopping timefor X . The latter assertion, in fact, holds except for countably many k’s, and henceWLG is assumed true. From the proof of Theorem 2.1 we have that X k (de�ned asin the proof of Theorem 2.1) is a solution to Eq. (1) up to the stopping time �k .From Theorem 10.2.2 of Stroock and Varadhan (1979), the linear growth conditionson b(·) and �(·) are su�cient for the non-explosion of X , and hence X is a solutionto Eq. (1), as k¿0 is arbitrary.

The following results show that under suitable conditions, the conditional momentsof the processes {Xn(t)} given Xn(0)= x converge as n→∞ to the correspondingmoments of the process X (t) given X (0)= x.


Proposition 2.6. Suppose that Eq. (1) admits a unique continuous solution; denotedby X . Assume also that there exists R¿0 such that for all i=1; : : : ; d and j=1; : : : ; r;b2i (x)+�

2ij(x)6R(1+‖x‖2), for all x∈Rd. Then under the conditions of Theorem 2:1,

for a �xed positive integer j and any t¿0; E([Xn(t)] j|Xn(0)= x) converges as n→∞to E([X (t)] j|X (0)= x) provided that �1 admits �nite absolute jth moments.

Proof. It follows from Theorem 2.1 that Xn de�ned by Eq. (4) converges weakly to Xas n→∞. It is easily seen from the Corollary to Theorem 25.12 of Billingsley (1986)that E([Xn(t∧�nk )] j|Xn(0)= x) converges as n→∞ to E([X (t∧�k)] j|X (0)= x), where�nk = inf{t¿0:‖Xn(t)‖¿k} and �k = inf{t¿0: ‖X (t)‖¿k}. From Example 5.3.15 ofKaratzas and Shreve (1991), E(‖Xt∧�k‖2)6C where C is a positive constant dependentonly on t, D and the initial point x0. The proof now follows directly from the Corollaryto Theorem 25.12 of Billingsley (1986).

3. Applications

3.1. The one dimensional case

It is well known that if b :R→R and � :R→R are locally bounded measurable func-tions and � is bounded below from zero, then Eq. (1) has a weak solution up to anexplosion time and this solution is unique in the sense of probability law. See Karatzasand Shreve (1991) for details. To ensure non-explosion we assume that there exists apositive function V ∈C2(R) and constants k1¿0; k2¿0 such that limx→∞ V (x)=∞and 1

2�2(x)V ′′(x) + b(x)V ′(x)6k1V (x) + k2 for all x∈R (see Stroock and Varadhan

(1979) for details). In particular the linear growth condition |b(x)|+ |�(x)|6k(1+ |x|),for all x∈R is su�cient for non-explosion. We also assume that b(·) and �(·) have�nite number of discontinuity points {a1; : : : ; ak} and de�ne for all i=0; · · · ; k + 1;fi(x)= x− ai; x∈R, where a0 =−∞ and ak+1 =∞. Then R=O1 ∪ · · · ∪Ok+1, whereOi= {x∈R :fi−1(x)¿0}∩ {x∈R :fi(x)¡0}; i=1; : : : ; k+1. Because �(·) is boundedbelow from zero and for each i=1; : : : ; k; (@fi(x)=@x)≡ 1, it then follows from Theo-rem 2.1 that Xn de�ned by (4) converges weakly to the unique solution X of (1) asn→∞. Moreover, if b and � satisfy the linear growth condition, Theorem 2.6 impliesthe convergence of conditional moments of Xn to those of X . The continuous-timethreshold (CTAR(1)) process, is de�ned as a solution to (1) with piecewise linear band piecewise constant �. Obviously the linear growth condition is satis�ed for anyCTAR(1) process. Hence, the results of Theorem 2.1, Propositions 2.5 and 2.6 holdfor any CTAR(1) process.

3.2. Continuous-time threshold autoregressive moving average (CTARMA)processes

A process {Y (t); t¿0} is a continuous-time linear ARMA(p; q) process with06q¡p and parameters (a1; : : : ; ap; b1; : : : ; bq; �; c) if

Y (t)= [1 b1 · · · bp−1]X(t); (9)


where

XT(t)= [X (t) X (1)(t) : : : X (p−1)(t)];

is a solution of Eq. (1) with

b(X(t)) =

0 1 0 · · · 0

0 0 1 · · · 0...

......

. . ....

0 0 0 · · · 1

−ap −ap−1 −ap−2 · · · −a1

X(t) +

0

0...

0

c

;

�(X(t))≡

0

0...

0

�

; (10)

where X (i)(t) is the ith derivative of X; �¿0; bq 6=0 and bj := 0 for j¿q. Weshall also write X(t)= (X1(t); : : : ; Xp(t))T. The CTARMA process is de�ned as a so-lution to Eqs. (9) and (10) where the parameters a1; : : : ; ap, c and � are constantover each of the l regions de�ned by ri−16 (d1; d2; : : : ; dp)X(t)¡ri, i=1; : : : ; l with−∞= r0¡r1¡ · · ·¡rl=∞, and the d’s are constants. For further discussions ofCTARMA models, see Brockwell (1994) and Stramer et al. (1996b). See Brockwell(1994), Stramer et al. (1996a,b) and Brockwell and Williams (1995) for conditionssu�cient for the CTARMA model to admit a unique solution. Consistency of theEuler scheme for simulating a CTARMA process has been shown by Stramer (1996)and Brockwell and Williams (1995) under restrictive conditions. We now generalizethe preceding results by showing the consistency of the Euler scheme under the as-sumption that X is the unique (in law) solution to Eq. (10) with piecewise-constantparameters. Now Xn de�ned by Eq. (4) becomes

Xn(t + n−1)=Xn(t) + n−1b(Xn(t)) + n−1=2�(Xn(t))�t ; t=0; 1=n; 2=n; : : : 1;

where �t are iid, of zero mean and unit variance, and Xn(t)=Xn([t=n]); ∀06t61. First,assume that dp 6=0 and de�ne for all i=0; : : : ; l; fi(x)=

∑pj=1 djxj− ri; x∈Rp. Then

Rp=O1 ∪ · · · ∪ Ol, where Oi= {x∈Rp: fi−1(x)¿0}∩ {x∈R: fi(x)¡0}, i=1; : : : ; l.Clearly �pp(·) and @fi(·)=@xp; i=1; : : : ; l are bounded below from zero. The weakconvergence of Xn→X then follows from Theorem 2.1.Next, consider the case when dp=0. Without loss of generality, assume dp−1 6=0.

Let K¿0. We shall show by adapting an argument in Brockwell and Williams (1995)that the amount of time that X spends in {x∈Rp: ‖x‖6K; |∑p−1

i=1 dixi|¡�} is “small”w.r.t �Leb, the Lebesgue measure. Let g be as de�ned in the proof of Lemma 2.2, and


�K = inf{t¿0: |X(t)|¿K}. We have

E

[p−1∑i=1

diXi+1(t ∧ �K)g′(p−1∑i=1

diXi(t ∧ �K))

−p−1∑i=1

diXi+1(0)g′(p−1∑i=1

diXi(0)

)]

=E

∫ t∧�K

0g′′(p−1∑i=1

diXi(s)

)(p−1∑i=1

diXi+1(s)

)2ds

+E

[∫ t∧�K

0g′(p−1∑i=1

diXi(s)

)S(s) ds

];

where S(s)=∑p−2

i=1 diXi+2(s) + dp−1(−ap(X(s))X1(s)− · · · − a1(X(s))Xp + c(X(s))).Hence, C�¿�1=2E[

∫ t∧�K0 5[−�; �](

∑p−1i=1 diXi(s))5[�1=2 ;∞)(

∑p−1i=1 diXi+1(s))

2 ds]; where C isa constant dependent only on ai(·); i=1; : : : ; p; c(·); t and K . This yields

E

[∫ t∧�K

05[−�; �]

(p−1∑i=1

diXi(s)

)ds

]

6E

[�Leb

{06s6t ∧ �K :

∣∣∣∣∣p−1∑i=1

diXi+1(s)

∣∣∣∣∣¡�1=4}]

+ C�1=2

=O(�1=4) + O(�1=2)=O(�1=4);

because it follows from the case with the coe�cient of Xp 6=0 that the �rst term onthe right hand side of the inequality is equal to O(�1=4). It can be similarly shown thatuniformly for all positive integer n,

E

[∫ t∧�K

05[−�; �]

(p−1∑i=1

diX in(s)

)ds

]=O(�1=4):

Thus, condition (A2) holds and hence the consistency of the Euler scheme. This com-pletes the proof.It is interesting to note that the above proof and Proposition 2.5 ensures that the

stochastic di�erential equation of a CTARMA model always admits a solution whichwe conjecture is unique.

3.3. Multidimensional AR(1) processes

Multivariate CTAR(1) process is de�ned in Stramer et al. (1996b) to be the unique(in law) solution of the non-linear di�erential equation (1) with b(X (t))=A(X (t))X (t) dt + c(X (t)) dt, where we assume that Rd is divided up into �nitely many poly-hedra O1; : : : ; Ok such that Rd=

⋃ki=1 Oi, the O’s have pairwise disjoint interiors, and

in each Oi, the d× 1 vector c(x) and the d×d matrices A(x); �(x) are constants. Wealso assume that a= ��T is positive de�nite in the interior of each Oi. Under theseconditions, Bass and Pardoux (1987) has shown that there exists a unique solution toEq. (1). Moreover, the conditions of Theorem 2.1 are satis�ed. Therefore, the Euler


scheme will always work in this case if the conditions of Theorem 2.1 on the frl’sare valid.

4. Conclusion

It is not hard to verify that using the Girsanov formula, the conclusion ofTheorem 2.1 holds for the case of a constant � and a bounded measurable b( ). Thissuggests that the conditions on the drift term in Theorem 2.1 may be relaxed. Theconvergence rates of the Euler scheme has been known under the classical case of con-tinuous coe�cients; see, e.g., Kurtz and Protter (1991b), Kloeden and Eckhard (1992).It is of great interest to investigate the rate of convergence of the Euler scheme whenthe coe�cients in Eq. (1) are discontinuous.

Acknowledgements

Research of K.S. Chan and O. Stramer was supported in part by NSF Grant DMS-9504798. The authors gratefully acknowledge their thanks to Tom Kurtz for his manyinsightful suggestions leading to a much improved version of the paper, and in par-ticular, a simpli�ed proof of the main theorem. The authors also thank a referee forconstructive comments.

References

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Billingsley, P., 1968. Convergence of Probability Measures. Wiley, New York.Billingsley, P., 1986. Probability and Measure. Wiley, New York.Brockwell, P.J., 1993. Threshold ARMA processes in continuous time. In: Tong, H. (Ed.), Dimension,Estimations and Models. World Scienti�c, Singapore, pp. 170–181.

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Nisio, M., 1973. On the existence of solutions of stochastic di�erential equations. Osaka J. Math. 10,185–208.

Ozaki, T., 1985. Non-Linear Time Series Models and Dynamical Systems.Stramer, O., 1996. On the approximation of moments for continuous time threshold ARMA processes.J. Time Ser. Anal. 17, 189–202.

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Stroock, D.W., Varadhan, S.R.S., 1979. Multidimensional Di�usion Processes. Springer, Berlin.Tong, H., Yeung, I., 1991. Threshold autoregressive modelling in continuous time. Statistica Sinica 1,411–430.

weak consistency of the euler method for numerically solving stochastic differential equations with...

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